Nonasymptotic random matrix theory aims to estimate spectral statistics (such as the extreme eigenvalues) of rather general random matrix models in a quantitative fashion. Such results are often first established under Gaussian or sub-Gaussian assumptions, and much work is then devoted to extending such bounds to more general situations. In this talk I will discuss a very different perspective on such problems: under remarkably weak structural assumptions, one can show in a precise nonasymptotic manner that the behavior of random matrices is accurately captured by that of an associated Gaussian model, regardless of the behavior of the Gaussian model itself. When combined with recent developments in the understanding of Gaussian random matrices, this nonasymptotic universality principle yields a powerful "black box" tool for understanding the behavior of extremely general nonhomogeneous and non-Gaussian random matrix models. If time permits, I will discuss applications to random graphs, spiked models, sample covariance matrices, and/or free probability theory. (Based on joint work with Tatiana Brailovskaya.)

https://sites.google.com/view/paw-seminar/